The Spectral Measure of Certain Elements of the Complex Group Ring of a Wreath Product
The Spectral Measure of Certain Elements of the Complex Group Ring of a Wreath Product
复制标题
花环积复群环某些元素的光谱测量
DOI：
10.1023/a:1020381532489
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发表时间：
2001
影响因子：
0.5
通讯作者：
T. Schick
中科院分区：
文献类型：
--
作者：
Warren Dicks;T. Schick
我们使用初等方法来计算L2维的特征空间的马尔可夫算子的点灯组和推广这个操作的其他群体。特别是，我们给出了一个透明的解释谱措施的马尔可夫运营商的点灯组Grigorchuk和祖克发现，后来使用他们，连同林内尔和希克，产生一个反例的强版本的Atiyah猜想的范围L2-Betti号码。我们用我们的结果来构造流形与某些L2-Betti数（作为收敛的有理数的无穷和），这是不明显的理性，但我们一直无法确定他们是否是不合理的。
We use elementary methods to compute the L2-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk and Zuk, and later used by them, together with Linnell and Schick, to produce a counterexample to a strong version of the Atiyah conjecture about the range of L2-Betti numbers. We use our results to construct manifolds with certain L2-Betti numbers (given as convergent infinite sums of rational numbers) which are not obviously rational, but we have been unable to determine whether any of them are irrational.